Noncommutative Kahler Geometry of the Standard Podles Sphere

TL;DR

This paper explores how classical Kahler geometry concepts like Hodge and Lefschetz decompositions, as well as Kahler identities, extend to the noncommutative setting of the standard Podles sphere under q-deformation.

Contribution

It introduces noncommutative analogs of key Kahler geometric structures and cohomologies for the Podles sphere, advancing noncommutative complex geometry understanding.

Findings

Noncommutative Hodge decomposition established

Lefschetz decomposition adapted to noncommutative setting

Kahler identities verified in the noncommutative context

Abstract

Building on the now established presentation of the standard Podles sphere as an example of a noncommutative complex structure, we investigate how its classical Kahler geometry behaves under $q$-deformation. Discussed are noncommutative versions of Hodge decomposition, Lefschetz decomposition, the Kahler identities, and the refinement of de Rham cohomology by Dolbeault cohomology